We shall use the results of Section . By the representation formula (REF ), we obtain that
\(u(t)&=\sum _{n=1}^{m}\langle u_0, \varphi _{n}\rangle E_{\alpha , 1}\left(-\lambda _{n} t^{\alpha }\right) \varphi _{n} + \sum _{n=m+1}^{\infty }\langle u_0, \varphi _{n}\rangle E_{\alpha , 1}\left(-\lambda _{n} t^{\alpha }\right) \varphi _{n} \\& =: u_1(t) + u_2(t). \) 

Observe that \(\Vert u(t)\Vert ^{2}=\Vert u_1(t)\Vert ^{2}+\Vert u_2(t)\Vert ^{2}\) . First, we have \(\lambda _n > 0\)  for \(n\ge m+1\)  and
\(\Vert u_2(t)\Vert ^{2}=\sum _{n=m+1}^{\infty }\langle u_0, \varphi _{n}\rangle ^{2} \left(E_{\alpha , 1} (-\lambda _{n} t^{\alpha })\right)^{2}.\) 
In Lemma REF , we have seen that the functions \(t \mapsto E_{\alpha , 1} (-\lambda _{n} t^{\alpha })\)  are completely on \([0,T]\)  for \(\lambda _n \ge 0\) . By [1]}, it follows that the functions \(t \mapsto \left(E_{\alpha , 1} (-\lambda _{n} t^{\alpha })\right)^{2}\)  and then \(t \mapsto \Vert u_2(t)\Vert ^2\)  are completely monotone on \([0,T]\) . Thus, Lemma REF  implies that the function \(t \mapsto \Vert u_2(t)\Vert ^2\)  is log-convex on \([0,T]\) . Therefore
\(\Vert u_2(t)\Vert  &\le \Vert u_2(0)\Vert ^{1-\frac{t}{T}}\Vert  u_2(T)\Vert ^{\frac{t}{T}} \\& \le \Vert u(0)\Vert ^{1-\frac{t}{T}}\Vert  u(T)\Vert ^{\frac{t}{T}}. \) 

On the other hand, \(\displaystyle \Vert u_1(t)\Vert ^{2}=\sum _{n=1}^{m}\langle u_0,\varphi _{n}\rangle ^{2} \left(E_{\alpha , 1}(-\lambda _{n} t^{\alpha })\right)^{2}\) .
It holds that the function \(\lambda \mapsto \left(E_{\alpha , 1} (-\lambda T^{\alpha })\right)^{2}\)  is non-increasing for \(\lambda \le 0\) . By \(\lambda _1 \le \lambda _n\)  for \(n\le m\) , we have
\(\Vert u_1(T)\Vert ^2 &\le \max _{1\le n \le m} \vert E_{\alpha ,1}(-\lambda _nT^{\alpha })\vert ^2 \sum _{n=1}^m \langle u_0, \varphi _{n}\rangle ^2\\& = \vert E_{\alpha ,1}(-\lambda _1T^{\alpha })\vert ^2 \Vert u_1(0)\Vert ^2\\&=: K_1^2 \Vert u_1(0)\Vert ^2.\) 

Since \(\lambda _n\le 0\)  for \(n\le m\) , the function \(t \mapsto \left(E_{\alpha , 1} (-\lambda _{n} t^{\alpha })\right)^{2}\)  is non-decreasing for \(n\le m\) . Hence, for \(0\le t\le T\) ,
\(\Vert u_1(t)\Vert  &\le \Vert u_1(T)\Vert  \le K_1\Vert u(0)\Vert ,\) 

where the constant satisfies \(K_1\ge 1\) . Thus
\(\Vert u_1(t)\Vert  &\le K_1 \Vert u(0)\Vert ^{1-\frac{t}{T}}\Vert u_1(T)\Vert ^{\frac{t}{T}}\\& \le K_1 \Vert u(0)\Vert ^{1-\frac{t}{T}}\Vert  u(T)\Vert ^{\frac{t}{T}}. \) 

From (REF ) and (REF ), setting \(K:= K_1+1\) , we obtain
\(\Vert u(t)\Vert  \le K\Vert u(0)\Vert ^{1-\frac{t}{T}}\Vert u(T)\Vert ^{\frac{t}{T}}, \qquad 0\le t\le T.\) 
